Thursday, December 27, 2012

Here’s a fun
math puzzle involving percents.The solution
is not terribly difficult – but it is surprising how many people end up getting
the answer wrong.Why not give it a try:

The owner of a farm stand puts a
bunch of watermelons on display at the beginning of the day.The watermelons have a mass of 200 kg.99% of the watermelons’ mass is water.It is hot, and during the day water evaporates
from the melons.At the end of the day, 98%
of the watermelons’ mass is water.Assuming that no watermelons were sold during the day, what is the mass
of the watermelons at the end of the day?

If you’d
like to check your answer, the correct solution is below.

If you’re
having a hard time, consider this puzzle. The math involved the same as the watermelon
problem, but the way in which the puzzle is presented tends to make the
solution easier to find.

Some guys planned a New Years’
party and invited only women.At the
party, there were a total of 200 people, 99% of whom were women.A lot of the women thought that the party was
lame, so they left.After one hour, 98%
of the people left at the party were women.How many total people were at the party after one hour?

This puzzle
is slightly easier to grasp because it explicitly states that there are men and
women.(And, it states that only women
leave the party.)

See below for the solutions.

Solution:
New Years’ party puzzle

It’s not
very hard to determine how many men and women there were at the beginning of
the party:

Women: 99% of 200 = 198Men:1% of 200 = 2

Since only women
left the party, there are still two men after one hour.However, these two men now comprise 2% of the
total party crowd.2 is 2% of 100…
therefore, there are 100 people at the party after one hour.

Solution:
Watermelon puzzle

The
watermelon puzzle is a little trickier simply because the problem doesn’t
explicitly state that the watermelon is made up of water and solids.Therefore, many people get hung up focusing
on the water.

If 99% of
the watermelon mass is water, then 1% is made up of solids.1% of 200kg is 2 kg – so there are 2 kg of
solids.Solids do not evaporate, so at
the end of the day, there are still 2 kg of solids.This now comprises 2% of the total watermelon
mass.2 kg is 2% of 100 kg.Therefore, there are 100 kg of watermelons
left at the end of the day.

Thursday, December 13, 2012

Riddles
and puzzles are a good way to challenge yourself and increase your logic
skills.They also give valuable practice
in solving the story problems that seem to haunt so many math students.

Here’s
a brain teaser that can be solved with simple math.(As is the case with many math puzzles, there
is more than one way to solve the problem.)If you think you’re up to it, try to answer the question before reading
the solutions and explanations at the bottom of this page.

There
are 340 players in a SET* tournament.Each game in the tournament is played with 4
people.For each game, there is one
winner who moves on to the next round.Rounds will continue until there is just one winner left – the champion.In any round, if the number of players is not
a multiple of 4, then some players will be chosen randomly to advance
automatically so that the number of people playing in that round is divisible
by 4.(This will ensure that each game
that is played will have 4 players.) How
many total games will be played in the tournament?

{Scroll down for solutions}

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*I
love card games.I also love activities
that help people learn and/or increase their brain power.SET is a card game that is simple to
learn, fun for a wide range of players (recommended for players 6 years old and
above), and helps build brain power.As
a teacher and card game enthusiast, I give SET my full recommendation.(And now SET-Junior is available for younger
players.)

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Solutions/Explanations

Method
1 (slow)

To
solve the problem, many people get out pencil and paper and begin calculating
the results, round by round.For each
round, they keep track of the number of games, and then add these up at the end
to find the answer to the question.This
method can be quite time consuming.One
such solution is shown:

Method
2 (fast)

Of
course, there is another way to solve this problem – a way that takes much less
time.In the previous solution, the
focus was on the WINNERS from each round.(The number of players in rounds 2 – 5 was determined by figuring out
how many winners there were in the previous round and adding this amount to the
number of people who were lucky enough to advance automatically.)A much quicker solution involves focusing on
LOSERS:

We
know each
game that is played will have 4 players.This means that each game is guaranteed to
have 3 losers.The tournament begins
with 340 players, and 339 of those players will be a loser at some point.All we need to do is divide 339 by 3 to find
our answer: There will be 113 games.

When
looking at a tough problem, we often hurt ourselves by focusing on the wrong
part of the problem.Sometimes it is
helpful to take a step back, reexamine the problem, and figure out if our focus
is in the right place.

Wednesday, December 5, 2012

We take
tests throughout our school years and most of us understand how the scoring
works.Take the number of points earned,
divide by the number of points possible, move the decimal a couple of spaces
right and, voila, we have a percent
that represents our score. (If you want to learn more about percents, there's a song for that.)

But then,
once a year, we take a standardized test and when the score is reported, we are
given a percentile.What in the world is a percentile?

Unfortunately,
we don’t learn much about percentiles in school.However, we do learn about the median, which
I discussed in my last blog entry, and also about quartiles.Let’s review briefly:

If all of
the values in a data set are arranged in order, then the middle number (or, if
there are two middle numbers, their average) is the median.This number divides the data set into two
equal groups:Half of the values in the
set are above the median, and half of the values are below the median.

If all of
the values in a data set are arranged in order, then the quartiles(there are three
of them) are the values that split the data set into four equal groups:

One-fourth
(25%) of the values in the set are less than or equal to the 1st
quartile, and three-fourths (75%) of the values are greater than or equal to
the 1st quartile.

Half (50%)
of the values in the set are less than or equal to the 2nd quartile,
and half (50%) of the values are greater than or equal to the 2nd
quartile.(The 2nd quartile
is the same as the median.)

Three-fourths
(75%) of the values in the set are less than or equal to the 3rd
quartile, and one-fourth (25%) of the values are greater than or equal to the 3rd
quartile.

If you
understand how quartiles work, then it’s not much of a leap to understand
percentiles.

If all of
the values in a data set are arranged in order, then the percentiles (there are 99
of them) are the values that split the data into 100 equal groups.(Note: some people call these centiles.)

1% of the
values in the set are less than or equal to the 1st percentile, and
99% of the values are greater than or equal to the 1st percentile.

2% of the
values in the set are less than or equal to the 2nd percentile, and
98% of the values are greater than or equal to the 2nd percentile.

And so on…

Because they
split the data set into so many groups, percentiles are only useful in
analyzing data sets that are very large – like the number of students who take
standardized tests.

If we
understand how percentiles work, then interpreting standardized test scores
becomes much easier.

If a score
is reported as 20th percentile, then 20% of test-takers scored at or
below that level.

If a score
is reported as 90th percentile, then 90% of test-takers scored at or
below that level.

That wasn’t
so hard was it?

So, why are
there so many people who struggle with percentiles?

I blame
politicians.

In the recent years, politicians arguing over
tax rates have consistently discussed “Americans in the top two percent of
earners”.They would do us all a great
service if they would instead argue about “Americans at or above the 98th
percentile of earnings”.It wouldn’t
make politicians any more likely to work well together, but at least people
would understand standardized test scores better!Affiliate Disclosure